Generalized Bagley-Torvik Equation and Fractional Oscillators

TL;DR

This paper generalizes the Bagley-Torvik equation by allowing fractional damping orders between one and two, analyzing the solution's properties, and revealing complex dependencies akin to thermodynamic phase transitions.

Contribution

It introduces a generalized fractional Bagley-Torvik equation and studies its solution, decay rate, and oscillation frequency, uncovering thermodynamic-like behaviors.

Findings

Solution expressed as convolution of trigonometric and exponential functions

Decay rate and frequency depend complexly on fractional order

Identifies thermodynamic-like phase transition properties

Abstract

In this paper the Bagley-Torvik Equation is considered with the order of the damping term allowed to range between one and two. The solution is found to be representable as a convolution of trigonometric and exponential functions with the driving force. The properties of the effective decay rate and the oscillation frequency with respect to the order of the fractional damping are also studied. It is found that the effective decay rate and oscillation frequency have a complex dependency on the order of the derivative of the damping term and exhibit properties one might expect of a thermodynamic Equation of state: critical point, phase change, and lambda transition.